The Enriched Xenon Observatory (EXO) will search for double beta decays of 136Xe. We report the results of a systematic study of trace concentrations of radioactive impurities in a wide range of raw materials and finished parts considered for use in the construction of EXO-200, the first stage of the EXO experimental program. Analysis techniques employed, and described here, include direct gamma counting, alpha counting, neutron activation analysis, and high-sensitivity mass spectrometry.